The Sonometer – The Resonant String and Timbre Change after pluckingPhysics of Music PHY103 Lab Manual The Sonometer – The Resonant String andTimbre Change after pluckingEQUIPMENT- Pasco sonometers (pick up 5 from teaching lab) and 5 kits to go with them- BK Precision function generators and Tenma oscilloscopes- Digital Tuners- Sets of wires for the sonometers (box of guitar strings with solderless leads attached). Theseare d’Addario strings EJ17 medium phosphor bronze and EXL120 extra light nickel.- Adaptors so output of sound sensors can be connected to preamps- Preamps connected to computers- Two of ¼” guitar cables are needed per set up so that the output of the sonometer detectors canbe put into the preamps and the output of the preamps can go into the audio input of thecomputers- BNC T-s so the sonometer detector signals can simultaneously be seen on the oscilloscope andrecorded- BNC male to male for the adapters- Rubber squares to prevent power out of the preamps going into the sonometer sensors- 2 kg weights per setup so that guitar like string tensions can be chosen.Sonometer sensor warning. Please keep the microphone power in the preamp off if you aresending the sonometer sensor signal through the preamps. One year some sonometer sensors weredamaged but we are not sure why.INTRODUCTIONMany musical instruments, such as guitars, pianos, and violins, operate by excitingoscillation modes in strings. This process has much in common with the excitation of modes in theair inside a tube (as we explore in other labs). The frequency of mode n for an ideal string that isfixed at both ends is given by the following formula:fn=nv2 L (Equation 1)This formula predicts a relationship between the length L of the string and the frequenciesfn at which the string will resonate. Here n is an integer. The lowest or fundamental mode wouldhave n=1. The velocity of the sound wave, v, is set by the tension and weight of the string. Thevelocity of waves along a string is given by the following equation: v =√Tρ (Equation2)In this equation, T is the tension in the wire and  is the linear density of the string. Thelinear density is the mass per unit length of the string.If you know the linear density of and tension on a string, you can calculate the frequencyat which the string will vibrate for a given length (like the length of a guitar neck or of a pianobackboard). In our case, we will mount the string on a sonometer. We can vary the tension on thestring and see how it changes the frequency at which the string will resonate.Physics of Music PHY103 Lab Manual The sonometer also allows us to “drive” the string into vibration (without plucking thestring) through the use of a magnetic field. This permits us to test which harmonics can themselvesbe made to ring within a certain length and type of string. The sonometer can also be used to seewhich parts of the string are vibrating. By replacing the velocity in equation 1 with that given inequation 2 we find the following: fn=n2 L√Tρ(Equation 3)Note that if T is in units of mass times acceleration or g m/s2 and ris in units of g/m then /T ris in units of m/s. This makes sense because it is a speed. If you then put L in units of m (meters)then fn (using the above formula) will be in units of Hz. The formula for the tension applied to thestring is approximately: T =(notch¿)mg ¿ (Equation 4)where m is the mass on the end of the string and g=9.81m/s is the gravitational acceleration.Notch number one is the notch closest to the sonometer string, while notch five is that farthest fromthe sonometer string. The sonometer comes with two rectangular bars. One is the wave driver and we connect that one tothe wave or function generator. This one will vibrate the string at the frequency set by the functiongenerator. The other bar is a sensor. It is like a guitar pickup. It senses the motion of the stringturning it into a voltage that can be seen with the oscilloscope or recorded into the computer usingthe microphone input to the computer and our microphone preamp.The above image shows a mode with n=3. Note that there are two nodes where the string is notmoving. Equation 1 predicts that this mode has 3 times the frequency of the fundamental and1/3 the wavelength of the fundamental or lowest mode. A node is where there is little motion;an antinode where there is maximal motion. The modes of oscillation have frequencies given by equation 1. Equation 1 shows that thefrequencies of the modes of oscillation are integer multiples of the lowest or fundamental frequency(that with n=1). fn = n f1 (Equation 5)The wavelength of the lowest or fundamental mode, λ1, is twice the length of the string. The wavelengths of the higher frequency modes are that of the longest one divided by n.λn = λ1/n (Equation 6)Each mode has n-1 nodes. 153sonometerPhysics of Music PHY103 Lab Manual Timbre changes: After a string is plucked many modes of oscillation are excited. However each mode or overtone may decay at a different rate leading to a change in the timbre of the sound. We can measure this affect by comparing the strength of one of the higher overtones to the fundamentalat different times after plucking. A string that has strong higher overtones at longer times after plucking is said to have a “bright” sound. A string with higher overtones that quickly decay after plucking is said to have a “soft” sound. The decay rate of overtones is affected by string composition, structure and tension. It is also affected by the stiffness of the bridge and nut and connections between bridge and nut and the rest of the instrument.PURPOSEOne purpose of this lab is to look at relationships between density, tension and frequencyin vibrating strings. We will try to use this relationship to understand some of the decisions madewhen choosing strings for different instruments. We also will explore how the string vibrates moreeasily at certain frequencies that we called modes. The spectrum of a plucked string will be relatedto its modes of vibrations. After a string is plucked different frequencies excited decay at differentrates. We can explore how the timbre of the note changes as the vibrations die away.The Pasco